Volovik effect in the ± s-wave state for the iron-based superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] M a y Volovik effect in the ± s-wave state for the iron-based superconductors Yunkyu Bang ∗ Department of Physics, Chonnam National University, Kwangju 500-757, Republic of Koreaand Asia Pacific Center for Theoretical Physics, Pohang 790-784, Republic of Korea (Dated: November 15, 2018)We studied the field dependencies of specific heat coefficient g ( H ) = lim T → C ( T , H ) / T and thermal con-ductivity coefficient lim T → k ( T , H ) / T of the ± s-wave state in the mixed state. We found that it is a genericfeature of the two band s-wave state with the unequal sizes of gaps, small D S and large D L , that Doppler shiftof the quasiparticle excitations (Volovik effect) creates a finite density of states, on the extended states outsideof vortex cores, proportional to H in contrast to the √ H dependence of the d-wave state. Impurity scatteringeffect on the ± s-wave state, however, makes this generic H -linear dependence sublinear approaching to the √ H behavior. Our calculations of lim T → k ( T , H ) / T successfully fit the experimental data of Ba(Fe − x Co x ) As with different Co-doping x by systematically varying the gap size ratio R = | D S | / | D L | . We also resolvethe dilemma of a substantial value of g ( H → ) but almost zero value of lim T → k ( T , H → ) / T , observed inexperiments. PACS numbers: 74.20,74.20-z,74.50
Introduction:
As to the pairing symmetry of the recentlydiscovered Fe-pnictide superconductors (SC) [1], the ± s-wave state [2, 3] is considered as the most natural and promis-ing pairing state. Genuinely being an s-wave gap state, thismodel is expected to provide exponentially activated behav-iors of various thermodynamic and transport properties dueto full gap(s) around Fermi surface(s) (FS) in the supercon-ducting phase. Angle resolved photoemission spectroscopy(ARPES) experiments [4] and early measurements of penetra-tion depth of M -1111 ( M =Pr, Nd, Sm) [5] and (Ba,K)Fe As [6] confirmed this expectation. However, some power laws oftemperature dependence measured, for example, with the nu-clear spin-lattice relaxation rate T − ∼ T [7] as well as thepenetration depth l ( T ) ∼ T − . on Ba(Fe,Co) As [8] werenot consistent with this simple picture. But these power lawdependencies were successfully explained theoretically withthe ± s-wave pairing state by the sign-changing feature of theorder parameter (OP) and the related noble impurity effect[9, 10] and hence strengthened the status of the ± s-wave stateas the pairing symmetry of Fe-pnictide superconductors.However, recent measurements of specific heat coefficient g ( H ) = C ( H ) / T and thermal conductivity coefficient k ( H ) / T in the mixed state and their field dependencies are posing anew challenge to the ± s-wave pairing model. Several mea-surements of specific heat coefficient showed a strong fielddependence: Ba . K . Fe As [11] ( (cid:181) H ), (Fe . Co . ) As [12] (sublinear in H ), and LaO . F . − d FeAs [13] ( (cid:181) √ H ).And thermal conductivity measured on Co-doped Ba-122 [14]and K-doped Ba-122[15] also showed various field dependen-cies. In particular, Tanatar et al. [14] measured k ( H ) / T ofBa(Fe − x Co x ) As for several values of x and found that thefield dependence of k ( H ) / T continuously evolves from expo-nentially flat, near linear in H , and to near √ H behavior withincreasing Co doping x .Another puzzling observation is that, while the zero fieldlimit of g ( H = ) / g normal in several experiments [11–13]showed a substantial values, the zero field limit of k ( H = ) / T [14, 15] of the same compounds with similar dopings approaches to negligibly small value. It implies that there ex-ist a large amount of zero energy excitations but they do notcontribute to thermal conductivity. Neither a nodal gap statenor a simple isotropic s-wave gap state can be easily compat-ible with these observations.In this paper, we study the field dependence of the spe-cific heat and thermal conductivity coefficients of the ± s-wavemodel in the mixed state using a semiclassical approximationwhich causes Doppler shift of quasiparticle excitations outsideof vortex cores (so-called ”Volovik effect” [16]). Recently,Mishra et al. [17] studied the field dependence of thermalconductivity on the isotropic ± s-wave model with equal sizegaps using a different method [18] and concluded that thismodel is incompatible with the experiments [14, 15]. Theirresult agrees with ours in that particular case. In the currentpaper, however, we show that it is essential to take into ac-count the unequal sizes of gaps in the ± s-wave model. Thisfeature is not only more realistic but also introduces genuinelynew physics to explain experimental data. We found that theunequal size of two isotropic s-wave gaps causes the field de-pendencies of k ( H ) / T as well as g ( H ) to be linear in H at thelow field regime.This generic H -linear dependence in the ± s-wave model isin contrast to the √ H dependence in the d-wave state [16] al-though both are due to Doppler shift effect on the extendedstates outside of vortex cores – it should also be contrasted tothe H -linear contribution to specific heat from the core boundstates in a single s-wave state [19] that cannot contribute tothermal conductivity. We then show that the impurity scatter-ing, combined with Volovik effect, make the generic H -lineardependence gradually modified to become sublinear and toapproach to the √ H dependence. Formalism:
For the generic two band model of the ± s-wavepairing state, we assume two isotropic s-wave order parame-ters (OPs) D S and D L of opposite signs on the two representa-tive bands of the Fe pnictide materials. D S and D L representa smaller and a larger size gap on the corresponding bands,respectively. All energy units in this paper is normalized by D L .We use the semiclassical approximation where the vectorfield term in Hamiltonian ec A ( r ) · k is replaced by the circulat-ing superflow velocity as v s ( r ) · k that causes Doppler shiftingof the quasiparticle excitations. The validity of this methodis discussed in length in Ref.[17] and it is a reliable approxi-mation from the low to intermediate field regime but could bequestionable approaching H c . Our results, therefore, shouldbe viewed with reservation at the high field regime where abetter method [18] should be applied in principle.The single-particle Green’s function of band a (= S , L ) inNambu matrix form including Doppler shift of the quasiparti-cle excitations is a function of r the distance from the vortexcore in addition to the usual momentum k and frequency w asfollows [16]. G a ( k , r , w ) = [ w + v s ( r ) · k ] t + x a ( k ) t + D a t [ w + v s ( r ) · k ] − x a ( k ) − D a (1)where t i are Pauli matrices and x a ( k ) is the quasiparticle en-ergy of band a . From the above Green’s function we obtainthe local density of states (DOS) of each band as N a ( w , H , r ) = − p TrIm (cid:229) k G a ( k , r , w ) . The Doppler shifting energy is givenas v s ( r ) · k = km a r cos q = b D L r cos q with normalized distance r = r / x ( x = coherence length) and ” b ” a constant of orderunity.We use the above Green’s function for 1 ≤ r ≤ R H / x withmagnetic length R H = a q F p H ( F a flux quanta, H magneticfield, and ” a ” geometric factor of order unity) because thecore region ( r <
1) is thermodynamically negligible if fieldsis not too close to H c . Eq.(1) shows that the gaps collapsewhenever the Doppler shifting energy v s ( r ) · k becomes largerthan the gap energy D a . The larger gap D L does not collapseeven with the largest Doppler shifting of v s ( r = ) · k F ≈ D L at the boundary of vortex core. But the small gap D S col-lapses for the region of 1 < r < b D L D S to create a finite DOS N S ( w = , H , r ) ≈ N normalS . With this observation, the mag-netic unit cell averaged DOS ¯ N a ( w , H ) = < N a ( w , H , r ) > cell = R R H x dr N a ( w , H , r ) / p R H is readily obtained at w = N L ( w = , H ) = p R H = N S ( w = , H ) = N normalS [( b D L D S ) − ] x R H (cid:181) H (3)Above Eq.(3) holds as far as D S < D L and shows thatVolovik effect immediately create a finite DOS with theisotropic ± s-wave state and there is no threshold value ofmagnetic field H ∗ to collapse the small gap D S . Its genericfield dependence is linear in H and its slope is proportionalto ≈ ( D L D S ) . We will show later with numerical calculationsthat impurity scattering will smoothen this generic linear-in- H field dependence and make it more sublinear and closer to (cid:181) √ H . Therefore the impurity effect is important to under-stand experiments.The impurity scattering is included by the T -matrixmethod, suitably generalized for the ± s-wave pairing model[10]. And impurity induced selfenergies renormalize the fre-quencies and OPs as w → ˜ w = w + S S ( w )+ S L ( w ) , and D S , L → ˜ D S , L = D S , L + S S ( w ) + S L ( w ) , with S , S , L ( w ) = G · T , S , L ( w ) where G = n imp p N tot ; n imp the impurity concentration and N tot = N S + N L is the total DOS. The T -matrices T , are the Paulimatrices t , components in the Nambu space. We refer read-ers to Ref.[10] for more details and it is straightforward toincorporate the local Green’s function Eq.(1) with Dopplershifting into the T -matrix method.After calculating the averaged ¯ N a ( w , H ) for all frequencies,specific heat is calculated as C a ( T , H ) = Z ¥ d w ¯ N a ( w , H ) w T sech ( w T ) (4)Similarly, thermal conductivity is calculated with [20] k a ( T , H , r ) (cid:181) N a v F Z ¥ d w w T sech ( w T ) K a ( w , T , H , r ) , (5) K a ( w , T , H , r ) = Im q ˜ z − ˜ D a × (cid:16) + | ˜ z | − | ˜ D a | | ˜ z − ˜ D a | (cid:17) (6)where ˜ z = ˜ w + v s ( r ) · k F . And then longitudi-nal and transversal thermal conductivities are cal-culated as k k ( T , H ) = R cell d r k ( T , H , r ) / p R H and k − ⊥ ( T , H ) = R cell d r k − ( T , H , r ) / p R H , respectively. Numerical results and discussions:
Since we consider twoband model, total specific heat and thermal conductivity arethe sum of two contributions from each band. In order to de-termine the precise contributions, we need to know the normalstate DOSs N S , L as well as Fermi velocity v F ( S , L ) of each band.However, in all numerical calculations in this paper, the largegap band is almost gapped even with a finite amount of impu-rity concentration so that its contribution to specific heat andthermal conductivity are very small compared to the contribu-tion from the small gap band. Therefore it is not very mean-ingful to determine the precise weighting of each band andwe simply put equal weighting to each band for convenientillustrations. For the field dependence of the OPs, we use aphenomenological formula D S , L ( H ) = D S , L p − H / H c .In Fig.1, we show numerical calculations of a pure case( G / D L = . | D S / D L | = . N S , L ( w , H ) of the small and large gap bands, respec-tively, as a function of magnetic field H . As expected, thesmall gap DOS ¯ N S ( w , H ) immediately starts collapsing withfield and reaches a constant value N smallS at w = H > . H c (see Fig.1(c)) while the larger gap DOS ¯ N L ( w , H ) remains gapped until H → H c . However, the gapped re-gion in ¯ N L ( w , H ) in frequency axis become extremely nar-row with increasing field which will affect specific heat coeffi-cient at high fields (see Fig.1(d))). Fig.1(c) shows normalized (b) N S ( , H ) H / H c L (a) N L ( , H ) H / H c L (d) N S , L , t o t ( = , H ) / N t o t no r m a l H/H c2 N tot ( ,H) N S ( ,H) N L ( ,H) (c) S , L , t o t ( H ) / t o t no r m a l H/H c2 tot (H) S (H) L (H) FIG. 1: (Color online) (a-b) Normalized DOSs with Doppler shiftingby magnetic field H , ¯ N S ( w , H ) and ¯ N L ( w , H ) for small gap and largegap bands, respectively, in clean limit ( G / D L = . w =
0, ¯ N S , L , tot ( w = , H ) / N normaltot of small gap andlarge gap bands, and the total, respectively. (d) Normalized specificheat coefficients g S , L , tot ( H ) / g normaltot of small gap and large gap bands,and the total contribution, respectively. zero energy DOSs ¯ N S , L ( w = , H ) and their total ¯ N tot ( , H ) = ¯ N S ( w = , H ) + ¯ N L ( w = , H ) . As explained above, beyond H ≈ . H c , the small gap ¯ N S ( w = , H ) reaches its full value( = . N tot ) and the large gap ¯ N L ( w = , H ) remains zero untilvery close to H c . Fig.1(d) shows normalized specific heat co-efficients g S , L , tot ( H ) , respectively. Some difference between¯ N L ( , H ) and g L ( H ) at the high field region is due to the finitetemperature (we used T / D L = /
50 to calculate the T → N L ( w , H ) at thehigh field region as seen in Fig.1(b).In Fig.2(a) and (b), we show calculations of total thermalconductivity and specific heat coefficients with varying gapsize ratio | D S / D L | = . , . , . , . , and 0.9, respectively.Here we used impurity scattering with G / D L = .
05 and uni-tary impurity ( c =
0) both for intra- and interband scatter-ing. These values are not particular choices for the plots. Awide range of impurity concentrations G and different strength ( H ) / N H/H c2 (b) [ pe r ,t o t ( H ) / T ] / [ N ,t o t / T ] H/H c2 s / l =0.2 s / l =0.3 s / l =0.5 s / l =0.7 s / l =0.9 (a) FIG. 2: (Color online) (a) Normalized total transverse thermal con-ductivity coefficient lim T → [ k ⊥ , tot ( H ) / T ] / [ k N , tot / T ] for differentgap size ratios, | D S / D L | = . , . , . , .
7, and 0 .
9, respectively. (b)Normalized specific heat coefficient g tot ( H ) / g tot , N for different gapsize ratios as in (a). All calculations include the same concentrationof impurities G / D L = .
05 with unitary scattering limit ( c =
0) bothfor intra- and interband scattering. ( c =
0) of impurity scattering produce qualitatively similar be-haviors. Fig.2(a) shows normalized total transverse thermalconductivity lim T → [ k ⊥ , S ( H ) / T + k ⊥ , L ( H ) / T ] / [ k normaltotal / T ] .Only the transverse k ⊥ / T (where the external field is appliedas H k c and the thermal current is measured in the plane as j th k ab ) are shown because those are the measured thermal cur-rent in most of experimental settings [14, 15]. The longitudi-nal thermal conductivity coefficient k k / T (where j th k c ) be-haves a bit more concave, in particular at the low field regime,because it corresponds to a parallel circuit of resistors whilethe transverse one corresponds to a series circuit. At thehigher field regime they become indistinguishably similar toeach other.The main feature of the results in Fig.2(a) is the systematicevolution of the slope of k ⊥ ( H ) / T at the low field limit. Fora large gap size ratio of | D S / D L | = .
9, lim H → k ⊥ ( H ) / T be-comes exponentially flat similar to the behavior of a single gapisotropic s-wave state. Then with decreasing gap size ratio of | D S / D L | , the low field slope quickly increases and the overall H dependence of k ⊥ ( H ) / T becomes concave down and closeto the √ H behavior for | D S / D L | = .
2. This concavity can bemade even stronger with decreasing the gap size ratio | D S / D L | and increasing impurity concentration G .Another important feature is that the values of k ⊥ ( H ) / T (also k k ( H ) / T although not shown here) in the zero field limitare negligibly small for all cases despite substantial impurityscattering; this is even true with the | D S / D L | = . ≈ √ H as in the d-wave case. In fact, theseextremely small values of the thermal conductivity coefficient k ⊥ ( H ) / T in the zero field limit was argued as an evidenceof an isotropic s-wave gap nature [14]. However, it is a verypuzzling feature when we note that the several experiments[11–13] observed substantial values of the specific heat co-efficients g ( H → ) with the same compounds with similardopings. Surprisingly, these seemingly conflicting behaviorsare also obtained in our theoretical calculations. In Fig.2(b),the normalized specific heat coefficient g ( H → ) / g tot , N , withthe same parameters as in Fig.2(a), show substantial values inthe zero field limit (0 . g tot , N is the full value of the small gapband contribution).It is a common knowledge that the DOS near Fermi levelsimilarly contributes to both specific heat and thermal con-ductivity. However, the discrepancy between g ( H → ) and k ( H → ) / T is possible due to the coherence factor of super-conductivity. The kernel of thermal conductivity (see Eq.(6)),being an energy current-energy current correlation function,has a destructive coherence factor (” − ” sign in the numera-tor of the last term in Eq.(6)) and the specific heat does nothave such coherence factor. In the case of d-wave pairing, thesame coherence factor becomes very weak in the low energylimit because the nodal gap D D ( q ) linearly disappears, so that g ( H → ) and k ( H → ) / T behave similarly.As to the field dependence of g ( H ) , with a relatively largegap size ratio, such as | D S / D L | = g ( H ) is verylinear in H for a substantial region of fields (up to ≈ H c / g ( H → ) . Decreasing the gap size ratio to | D S / D L | = g ( H ) be-comes gradually more concave down. This behavior is in ex-cellent agreement with the measurements of Ba . K . Fe As [11] ( ≈ H ), (Fe . Co . ) As [12] (sublinear in H ), andLaO . F . − d FeAs [13] ( ≈ √ H ). Conclusion:
Using semiclassical approximation, we havecalculated specific heat coefficient g ( H ) and thermal con-ductivity coefficient k ( H ) / T of the ± s-wave model in themixed state with including impurity scattering. We found thatDoppler shift on quasiparticle excitations on extended statesimmediately induce zero energy excitations proportional to H from a smaller gap band. There are no threshold values ofsmall gap size D ∗ S and magnetic field H ∗ to induce such zeroenergy excitations. This steep increase of g ( H ) and k ( H ) / Tlinear in H is a generic behavior of any two gap isotropics-wave superconductors with unequal sizes of gaps.
There-fore, our results also provide natural explanation for the ther-mal conductivity measurements in MgB [21] and NbSe [22].The sign-changing feature of the ± s-wave state is not a pri-mary source of the field dependence of g ( H ) and k ( H ) / T but plays an important role through impurity scattering. Theimpurity scattering effect with sign-changing OPs [10] ef-ficiently accumulate zero energy impurity band near Fermilevel and reflects it to g ( H → ) (see Fig.2(b)) and makes thefield dependence of k ( H ) / T and g ( H ) more concave downand smoothly curved.Having found substantial contribution of low energy exci-tations from extended state outside of the vortex cores, wejustified the ignoring of the core region contribution a pos-teriori . We compare the numerical calculations to the ther-mal conductivity measurement [14] of Ba(Fe − x Co x ) As andshowed that the systematic evolution of the thermal conduc- tivity coefficient k ( H ) / T with Co-doping x is well explainedby changing the gap size ratio | D S / D L | in our theoretical cal-culations, which is also consistent with the Fermi surface evo-lution in Ba(Fe − x Co x ) As with electron doping by Co in therigid band picture. We also showed that the conflicting behav-iors between g ( H ) and k ( H ) / T in the H → ± s-wave statewith isotropic s-wave gaps of unequal sizes can consistentlyexplain the field dependence of g ( H ) [11–13] and k ( H ) / T ob-served in Co-doped Ba-122 [14] and K-doped Ba-122 [15]. Acknowledgement –
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